Question 6 \& of $9 \operatorname{step} 3$ of 3 $00: 27: 51$ A parenting magazine reports that the average amount of wireless data used by teenagers each month is $10 \mathrm{~Gb}$. For her science fair project, Ella sets out to prove the magazine wrong. She claims that the mean among teenagers in her area is less than reported. Ella collects information from a simpte random sample of 16 teenagers at her high school, and calculates a mean of $9.2 \mathrm{~Gb}$ per month with a standard deviation of $1.9 \mathrm{~Gb}$ per month. Assume that the population distribution is approximately normal. Test Ella's claim at the 0.025 level of significance. Step 3 of 3: Draw a conclusion and interpret the decision. Answer 2 Points Tables Keypad
Solution
Step 1 ：First, we need to calculate the z-score. The z-score is calculated by subtracting the population mean from the sample mean, and then dividing by the standard deviation divided by the square root of the sample size.
Step 2 ：Given values are: population mean = 10, sample mean = 9.2, sample standard deviation = 1.9, sample size = 16. Using these values, the z-score is calculated as \( z = \frac{{9.2 - 10}}{{1.9 / \sqrt{16}}} = -1.6842105263157912 \)
Step 3 ：After calculating the z-score, we can use a z-table or a statistical function to find the p-value. The p-value is the probability that we would observe a result as extreme as our sample mean, if the null hypothesis were true.
Step 4 ：The calculated p-value is 0.04607049482694396
Step 5 ：We compare the p-value with the level of significance, which is 0.025 in this case. If the p-value is less than the level of significance, we reject the null hypothesis. If the p-value is greater than the level of significance, we fail to reject the null hypothesis.
Step 6 ：The p-value is greater than the level of significance. Therefore, we fail to reject the null hypothesis. This means that we do not have enough evidence to support Ella's claim that the mean amount of wireless data used by teenagers in her area is less than the reported average of 10 Gb per month.
Step 7 ：\(\boxed{\text{We fail to reject the null hypothesis. We do not have enough evidence to support Ella's claim.}}\)
